# National Plumbers

The National Plumbers skit is performed for logic and CS classes at some sites, usually on the last week of the session during evening period/study hall.

# Plot

## Introduction

The narrator does a monologue for this section.

Zeno is known for his paradox, but he was also the world's first plumber. However, he wasn't very good, and had very little value as a plumber. Whenever he got called to a job, he would make it halfway before having to stop and ask for directions. He made it half of the remaining distance before having to ask for more directions. Etcetera, etcetera.

To increase efficiency, he hired a supervisor (super). Business boomed. It boomed so much that Zeno's super hired a super, and that super hired a super. Eventually, there were many Plumbers around Ancient Greece. They decided to unionize into the National Plumbers Association (NPA).

## First Union Meeting

Guiseppe, the union manager, runs the first union meeting. Zeno is not present. However, his super, Juan Carlos, is present. So is Juan's super, Stuart, Stuart's super, Theresa, and Theresa's super, Florio (sometimes called Floris or Forrest). For everyone who is present, their super is present. Guiseppe concludes that all members except Zeno are here. He then lays down the five by-laws of the union:

1. Zeno is a National Plumber (NP).
2. Every NP has a super who is also an NP.
3. No NP has Zeno as a super.
4. No two NPs have the same super.
5. Given any setting, if Zeno is present in that setting, and any NP being in that setting means that their super is in that setting, than the entire NPA is in that setting.

yeah Guiseppe further declares that all meetings will be held at Zeno's house to satisfy the first condition of Law 5.

## Meanwhile...

Zeno is driving to the meeting. He forgets where he is going, sees a gas station, and asks for directions. The attendant gives him directions. Zeno tells him to "fill it up halfway".

This is repeated three or more times, with the same actor playing the attendant, but with a different costume. The gas station is a "family business".

## The Telephone Operator

The policy of the NPA is that a customer must call twice before a plumber is sent. The telephone operator explains how he determines who to send after a customer calls for two plumbers:

1. If a customer calls for any plumber, and then for Zeno, he sends the first plumber.
2. If a customer calls for any plumber, and then for the super of another plumber, he sends the super of who he would have sent had the customer called for the first plumber and then the second.

In shorthand,

1. X T Zeno = X
2. X T S(Y) = S(X T Y)

The operator uses an example:

Rudy Lily has a clogged sink. First, she calls for Juan, and no one is sent. A few hours later, she calls for Stuart, and the operator is ready to use his rules.

Juan T Stuart = Juan T S(S(Zeno)) = S(Juan T S(Zeno)) = S(S(Juan T Zeno)) = S(S(Juan)) = Theresa

by Substitution, Rule 2, Rule 2, Rule 1, and Substitution respectively.

He sends Theresa to Rudy Lily.

The operator then proves that X T S(Zeno) = S(X). However, he cannot prove that S(Zeno) T X = S(X), which seems to be always true.

## Break

The actors pass out a "Telephone Operator Licensing Exam" to practice using the TO-rules.

## Diophantus

The narrator introduces Diophantus, who can solve the operator's problem.

First, Diophantus uses the Five Laws to show that the Association was structured in a sort of line, starting at Zeno and continuing indefinitely.

Then, Diophantus uses Rule 5 to prove the operator's conjecture:

Let the setting include all NPs that satisfy the operator's conjecture. Zeno is there because, by Rule 1, S(Zeno) T Zeno = S(Zeno)

Let a generic NP named Mario be in the setting. Then, S(Zeno) T Mario = Super(Mario). S(S(Zeno) T Mario) = S(S(Mario)). S(Zeno) T S(Mario) = S(S(Mario)). Thus, S(Mario) is in the setting.

ALL NPs are in the setting.

## Reveal

The narrator then reveals the true meaning of the National Plumbers. At LAN.18.2, some inattentive students resulted in the reveal being canceled.